Existence conditions of formal first integrals of Lorenz system

Abstract

In the present paper, we study the formal first integrals of the well-known Lorenz system. First, for the simple condition of 0 = s we consider the form of the first integral. Furthermore, when 0 ≠ s and b is not a negative rational number, the Lorenz system dose not have any nontrivial formal first integral in a neighborhood of the origin.